Potential theory of subordinate killed Brownian motion in a domain

Subordination of a killed Brownian motion in a bounded domain D ⊃ ℝ^d via an α/2-stable subordinator gives a process Z_t whose infinitesimal generator is -(-Δ|_D)^α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Z_t in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the semigroup of Z_t, and, in the case when D is a bounded C^1,1 domain, obtain bounds on the Green function and the jumping kernel of Z_t.